

The complete proof is omitted due to space limitations. Essentially, proposition REF  can be proven by using Lemma REF , the convergence result with a quantized model update [1]}, and replacing the SGD weight update in [2]} with (REF ).

From Proposition REF , We let (REF ) be upper bounded by \(\epsilon \)  in () as follows
\(\mathbb {E}[F(w_T) - F(w^*)] \le \frac{L}{2}\frac{v}{T+\gamma } \le \epsilon . \) 
